Review question

For which $0^\circ < \theta < 500^\circ$ is $3\cos \theta + 4\sin \theta > 2$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8709

Suggestion

$3\cos \theta + 4\sin \theta$ may be written in the form $r\cos(\theta - \alpha)$, where $r > 0$.

1. Calculate the value of $r$, and show that one value of $\alpha$ is approximately $53.1^\circ$.

Can we expand the expression $\cos(\theta - \alpha)$?

1. Hence show that one solution of $3\cos \theta + 4\sin \theta = 2$ is approximately $120^\circ$.

Can we use the result from part (a)?

1. State all the other solutions for $0^\circ \leq \theta \leq 500^\circ.$

How many solutions do you expect in the interval $0\leq\theta<360^\circ$?

How can we locate further solutions?

1. Hence give the positive values of $\theta$ less than $500^\circ$ for which $3\cos \theta + 4\sin \theta > 2$.

A sketch graph of $y = 3\cos \theta + 4\sin \theta$ might be helpful. What happens at each of the solutions found in part (c)?